Chapter 6

• Failure resulting from fluctuating

load

Fluctuating load?

What is special about it?

1 M S Dasgupta BITS Pilani

• Variable loading results when the applied load or the induced stress on a component is not constant but changes with time

• In reality most mechanical components experience variable loading due to

-Change in the magnitude of applied load

-Change in direction of load application

-Change in point of load application

Fluctuating / Variable load

2

Stress variation: Sinusoidal

2stresseor variabl amplitude

2stressmean or midrange

stress of range

stress maximum

stress minimum

minmax

minmax

minmax

max

min

a

m

r

Completely Reversed : mean stress is zero; equal reversals on both sides; useful in conducting experiments

Repeated stress: minimum stress is zero; mean stress equal to half of the range stress

Fluctuating stress: maximum, minimum and mean stress are all non-zero and arbitrary

Idealized types of cyclic loading: Sinosoidal

Fatigue

• Fatigue is a phenomenon associated with

variable loading or more precisely to cyclic

stressing or straining of a material

• ASTM Definition of fatigue

– The process of progressive localized

permanent structural changes occurring in a

material subjected to conditions that produce

fluctuating stresses at some point or points

and that may result in cracks or complete

fracture after a sufficient number of

fluctuations. 5

Fatigue failure in Metals

Crack initiation at

the outer surface

Beach marks

showing the nature

of crack propagation

Final rupture occurs

over a limited area,

characterizing a very

small load required

to cause it

Crack initiation, propagation and rupture in a shaft subjected to repeated bending

6

Fatigue Life Prediction

predict the failure in number of cycles N to failure for a specific type of

loading 33 10 :(HCF) fatigue cycleHigh ;101:(LCF) fatigue cycle Low NN

• Stress life methods

– Based on stress levels only

– Least accurate of the three, particularly for LCF

– It is the most traditional because easiest to implement for a wide range of applications

– Has ample supporting data

– Represents high cycle fatigue adequately

• Strain life methods

– Involves more detailed analysis of plastic deformation at localized regions

– Good for LCF

– Some uncertainties may exist in results because several idealizations get compounded

– Hence normally not used in regular (special occasions)

• Linear elastic fracture mechanics methods (LEFM)

– Assumes that crack is already present and detected

– The crack location is then employed to predict crack growth and sudden rupture with respect to the stress nature and intensity

7

The S-N Diagram for steel (UNS G41300), normalized, Sut=812 MPa.

Endurance Limit,

It is the stress at which the

component can sustain

infinite number of cycles

S-N Diagram

R. R. Moore high-

speed rotating

beam machine.

Non-Ferrous materials tested up to 5*108 cycles

S’e

8

Sut – S’e relation

conditions loading actual in thelimit Endurance

bending reversein obtainedlimit Endurance

1460700

146050

'

'

e

e

ut

utut

e

S

S

MPa Sfor MPa

MPaSfor S.S

9

Se S’e relation

'

eedcbae SkkkkkS

factoronmodificatieffects ousmiscellane

factory reliabilit

factoron modificati etemperatur

factoron modificati load

factoron modificati size

factoron modificaticondition surface

f

e

d

c

b

a

k

k

k

k

k

k

10

b

uta aSk

Surface cond. Mod. factor (ka)

The surface modification factor depends on the quality of the

finish of the actual part surface and on the tensile strength of

the part material.

Size modification factor, kb

1. effect, size no loading axialFor

25451000837.0859.0

5179.224.162.7/

:only torsion and bendingin barscircular rotatingFor

107.0107.0

b

b

k

mmdifd

mmdifddk

Concept of Equivalent Diameter de

What happens when bars are not rotating but

say under bending.

Or non-circular bars like square, or I section?

Kb for non-rotating shapes

Effective dimension “de”

obtained by equating the

volume of material stressed

at and above 95 percent of

the maximum stress to the

same volume in the

rotating-beam specimen

Load modification factor, kc

torsion

axial

bending

kc

,59.0

,85.0

,1

Actually the kc is sensitive

to Sut of the material. Tables

6-11 to 6-14 (page no. 333)

in Text Book give the

details. The above values

are representative.

Temperature modifying factor, kd

Brittle fracture is a strong possibility when

operating temp is below RT

At temp. higher than RT, yielding should be

investigated first because the yield strength drops

off rapidly with temperature.

Creep at elevated temperature

Temperature modifying factor, kd

FT

where

TTTTk

o

F

FFFFd

100070

10595.010104.010115.010432.0975.0 41238253

For carbon and alloy steels experimental result

expressed as a fourth-order polynomial curve fit

to the data underlying

Or interpolate from a chart / table of

operating temp. vs tensile

ae zk 08.01

Reliability factor, ke

Based on standard

deviation of Endurance

strength data

Accounts for

– Residual stress

– Coating failure

– Frettage corrosion material of mating part.

– Synergic effect of corrosion and temperature

where is Se is function of frequency of loading.

Miscellaneous effects factor, kf

Kf is a reduced value of Kt and it is also called fatigue

strength reduction factor

Actual / Fatigue stress concentration factor, Kf

factor) (geometricfactor ionconcentrat stress lTheoretica

21)-6& 20-6 Fig.(from y valuesensitivit notch

tK

q

1111 tsshearfstf KqKorKqK

specimen free-notch in stress

specimen notched in stress maximumfK

Stress-concentration factors for a variety of geometries under

different loading conditions can be found in Table A–15

(page:1026-1032) 19

Notch Sensitivity

20

21

Estimation of Kf

Kf = 1+q(Kt -1).

•When q=0, the material has no sensitivity to notches, Kf=1.

•When q=1, or when notch radius is large for which q is

almost equal to 1, the material has full notch sensitivity, and

Kf = Kt.

•For all grades of cast iron, use q=0.20.

•Use the different graphs to obtain q for bending/axial and

torsional loading.

•Whenever the graphs do not give values of q for certain

combinations of data, use either Neuber equation or

Heywood equation.

22

Use the Neuber equation when the notch is circular/cylindrical.

Estimation of Kf

radiusnotch

strength. ultimate offunction i.e ),(

constant material a is andconstant Neuber is a where

11

1

1

r

Sfa

KqKand

r

aq

ut

tf

For steel, with Sut in kpsi, the Neuber constant can be

approximated by a third-order polynomial fit of data as

38253

38253

)10(67.2)10(35.1)10(51.219.0:

)10(67.2)10(51.1)10(08.3246.0:

ututut

ututut

SSSaTorsion

SSSaaxialorBending

100p

si =

0.6

89M

Pa

23

Use Heywood equation when the notch is NOT circular/cylindrical but is a

tranverse hole or shoulder or groove.

Estimation of Kf

335 page 15;-6 Table thein given are values

121

a

where

r

a

K

K

KK

t

t

tf

r= hole/ shoulder/groove size

24

Sn’

Goodman Method

m

a

-Sy

Sy

Sy Su

FATIGUE

FAILURE REGION

NO FATIGUE

FAILURE REGION

Goodman Line

0

Yield Line (Langer line)

1

u

m

n

a

SS

Predictor of failure in ductile materials

experiencing fluctuating stress

Sn’ = endurance strength

a = alternating stress

m = mean stress

25

Sn’/N

Sn’

Goodman Diagram

m

a

-Sy

Sy

Sy Su

FATIGUE

FAILURE REGION

Goodman Line

0

Yield Line

1

u

m

n

a

SS

Su/N

NSS u

m

n

a 1

Safe Stress Line

Safe Stress Line

SAFE ZONE

Sn’ =endurance strength

a = alternating stress

m = mean stress

26

Design under cyclic loading

fyt

m

e

a

nSS

1

1yt

m

e

a

SS

1

ut

m

e

a

SS

1

2

ut

m

e

a

SS

1

22

yt

m

e

a

SS

27

28

Different fatigue failure models

yielding)staticfor

checkingfor (only lineLanger 1

line EllipticASME1

lineGerber 1

line Goodman Modified1

line Soderberg1

222

2

yyt

m

yt

a

fyt

m

e

a

fut

mf

e

a

fut

m

e

a

fyt

m

e

a

nSS

nSS

nSn

S

nSS

nSS

M S Dasgupta BITS Pilani 29

Modified Goodman and

Langer Failure Criteria

Important Intersections in First Quadrant

30

Important Intersections in First Quadrant

Gerber and Langer

Failure Criteria

31

Important Intersections in First Quadrant

ASME-Elliptic and Langer

Failure Criteria